The terms sueing and lawsuit are the verb and noun describing the same legal action.
If you sue someone, you’re accusing them (in court) of doing something wrong or illegal and demanding that they pay for it. Almost every time someone sues, they’re looking for money. For example, “Dan sued Leslie for injuring him in an accident”.
The term lawsuit is a noun that refers to the legal action of sueing someone. For example, “Dan filed a lawsuit against Leslie for injuring him in an accident.”